Comparative Study on Hydrodynamic Characteristics of Under-Water Vehicles Near Free Surface and Near Ice Surface

Abstract

In this paper, the commercial computational fluid dynamics software STAR-CCM+ (18.04.008-R8) is utilized to analyze the hydrodynamic performance of BB2 underwater vehicles under various navigation conditions, as well as the flow field disturbances caused by the free surface and ice surface during navigation. After dividing the computational domains based on different navigation scenarios, numerical simulations are conducted for BB2 underwater vehicles (without a propeller) at infinite depth, near the free surface, and near the ice surface under various operating conditions. The analysis focuses on changes in resistance, velocity fields, and pressure fields of the BB2 at different velocities and navigation depths, followed by a comparison of the navigation differences of BB2 vehicles under varying operating conditions. Furthermore, to simulate realistic navigation conditions for underwater vehicles, numerical simulations are performed for BB2 underwater vehicles equipped with a propeller under different operating conditions. The results indicate that both the free surface and ice surface significantly influence the resistance, velocity field, and pressure field of the BB2. When the navigation depth exceeds 2D, the impact of ice on the vehicle can be nearly disregarded, and when the navigation depth exceeds 3D, the influence of the free surface on the vehicle can also be considered negligible.

1. Introduction

The 21st century is the century of the ocean. The intensification of the greenhouse effect and the accelerated melting of ice and snow in polar regions have not only resulted in significant changes in polar ecological environments but have also presented both opportunities and challenges for human exploration in these areas. The rapid development of underwater vehicles has facilitated operations in hazardous environments. These underwater vehicles enable researchers to explore the depths of the ocean and access polar regions that were previously difficult for humans to reach, highlighting the growing recognition of their importance. The “REMUS” unmanned underwater vehicle [1] was developed by the United States as early as 1998. In 2007, the Woods Hole Oceanographic Institution (WHOI) in the United States developed the “SeaBED series AUV” with a catamaran structure [2] and used “PUMA AUV” and “JAGUAR AUV” series for the detection of subglacial deep-sea hydrothermal plume and seabed surveying and mapping in summer. In 1997, the United Kingdom produced an AUV [3] specifically for Marine exploration and subsequently used the AUV for military research. In addition, the Canadian “Theseus AUV” and “Arctic Explore AUV” [4] were successfully used for the laying of polar optical cables and subglacial surveying and mapping in 1996 and 2010, respectively.

In recent years, the research on polar research equipment in China has witnessed rapid development. A novel autonomous/remote-controlled hybrid underwater robot, “polar ARV”, independently developed by the Shenyang Institute of Automation, Chinese Academy of Sciences [5], was successively applied for the continuous observation of designated sea ice areas in China’s third, fourth, and sixth Arctic scientific expeditions from 2008 to 2014. On 3 January 2020, the “Discovery 1000” underwater robot was applied in China’s 36th Antarctic Scientific Expedition [6,7], which provided important technical support for the expedition team in carrying out a comprehensive survey of multiple environmental elements in the Ross Sea.

At present, a lot of research work has been conducted on the model of vehicles without a propeller at home and abroad. Tolliver et al. [8] found that the near-surface navigation of underwater vehicles is affected by tail suction, which can cause the vehicle’s nose to become buried; therefore, during near-surface operations, it is essential to accurately control the depth or velocity of the vehicle to minimize the impact on the water’s surface. Sahin et al. [9] predicted the force and moment coefficients of the AUVs with single and fin configurations. Jagadeeshd et al. [10] first studied the changes in the lift and pitch torque coefficients of AUV at different velocities and angles of attack at a large diving depth in a towing pool, and they [11] further discussed the influence of the free surface on the axisymmetric rotating body model with RANS using Fluent through a combination of experimental and numerical simulations. By comparing different Froude numbers and different dimensionless depths, Shariati et al. [12] compared the hydrodynamic characteristics of a bare hull and appended hull and discussed the influence of the appendages on the hydrodynamic characteristics of the submarine. After verifying the feasibility of numerical calculation methods, Wang et al. [13] discussed the near-surface effects on the hydrodynamic performance of submarines. Bai et al. [14] used the RANS method in the STAR-CCM+ (15.02.007-R8) in combination with the SST k-ω turbulence model to study the hydrodynamic properties of the hull near the ice surface. With the adoption of the axisymmetric bare hull model, Luo et al. [15] selected the axisymmetric bare boat body model (without a propeller) and, based on VOF-free surface capture technology, conducted a comparative analysis of the hydrodynamic performance of UUV under different sailing conditions, and compared the eddy current field changes in UUV under different operating condition. Wang et al. [16] conducted numerical research on the three-dimensional viscous flow field of the Suboff main hull model based on STAR-CCM+ (15.02.007-R8) simulation software and analyzed the influence of different turbulence models and grid divisions on the calculation results. Liu et al. [17] used the large eddy simulation (LES) method to study the control mechanism of a vortex control baffle for the horseshoe vortex around the sail of a DARPA SUBOFF model. Hao et al. [18] developed a deep graph learning model to simulate the 2D wake flows of the DARPA SUBOF without a propeller.

In order to accurately simulate vehicle navigation under various operating conditions, many researchers have increasingly shifted their focus to the self-propulsion of vehicles, following extensive experiments and numerical simulations of vehicle models without a propeller. The early self-propulsion of vehicles was based on the geometrical shapes of DARPA Suboff and ONR Body1. Liefvendahl and Troëng [19], Chase and Carrica [20], and Sezen et al. [21] introduced the research on the self-propulsion of DARPA Suboffice with an E1619 propeller. Chase and Carrica [20] determined the operating conditions applicable to different turbulence models, which provided a reference for the calculation of the vehicle’s self-propulsion. Zhang et al. [22] studied the mutual interference between the hull and propeller near the free surface, and the results showed that the influence of the free surface on resistance was greater than that of the self-propulsion factor. Özden et al. [23] conducted a comprehensive study regarding the experiments and simulation on the same model as Chase et al. [20] and provided self-propulsion test data of Suboff submarines with E1619 propellers. Wang [24] carried out tests and simulations of general submarine-propeller wake near the water surface, and the results showed the interaction between the hull and the free surface has a significant impact on the inflow and wake of the propeller, resulting in a higher local propulsion coefficients and blade load near the surface of the propeller. With the adoption of the CFD method, Li et al. [25] carried out numerical simulation calculations of the submarine’s self-propulsion model at two diving depths and three velocities using the URANS equation and analyzed the changes in oncoming flow and hydrodynamic load on the propeller. Adrian [26] performed a calculation on the Suboff model appended with an E1619 paddle near the free surface, which emphasized the influence of the free surface on the propulsion performance loss of navigation. Dongjin et al. [27] established a 1/15 BB2 model and conducted a full-size resistance and propulsion test in the towing pool of the Korea Research Institute of Ships and Ocean Engineering (KRISO), finding that when the diving depth is about three times the diameter of the submarine, the free surface effect is negligible. Ding et al. [28] conducted numerical studies on the hydrodynamic characteristics and self-propulsion performance of DARPA Suboff under wave–current coupling conditions. It was found that under the coupled submarine-generated waves and wave–current effects, when the submarine approaches the free surface, the wave heights and wavelengths significantly impact the self-propulsion performance.

So far, extensive research has been conducted on the navigation of vehicles at significant diving depths and near the free surface; however, there are few studies focused on vehicles navigating near the ice surface, the influence of changes in the flow field, and the effects of appendages on underwater vehicles. Additionally, research and analysis of self-navigation for underwater vehicles in various environments are also scarce. This study investigates the BB2 model equipped with a tail wing. Utilizing computational fluid dynamics (CFD), the relative motion of the underwater vehicle is simulated under various operational conditions by maintaining the model in a fixed position while applying an incoming flow velocity; no degrees of freedom are open. Initially, numerical simulations were conducted for the BB2 underwater vehicle (without a propeller) across different operational scenarios. The necessity of appendages in this analysis is discussed, and the variations in resistance, velocity field, and pressure field of the BB2 underwater vehicle are analyzed at different velocities and depths. Furthermore, a comparative analysis of the navigation characteristics of the BB2 vehicle under diverse operational conditions is presented. Subsequently, to more accurately replicate the actual navigation of the underwater vehicle, numerical simulations of the BB2 model (with a propeller) were performed under varying operational conditions. A comparative analysis of the changes in the flow field across these different conditions was undertaken, leading to relevant conclusions.

2. Research Model

The parameters of the BB2 model without a propeller are shown in

Table 1. Main dimensions of BB2.

Overall Length [m]	1.679
Diameter (D) [m]	0.23
Truncated Model Length (LM) [m]	1.607
Star of parallel midbody aft of leading edge [m]	0.383
End of parallel midbody aft of leading edge [m]	1.149
Wetted Surface Area (S) [m2]	1.041
Hull Prismatic Coefficient	0.785

, and the three-dimensional diagram of the model is shown in Figure 1. In this study, Section 3, Section 4 and Section 5 all use the research model shown in Figure 1. The ratio of the distance from the centerline of the BB2 model to the free surface and the ice surface (H) to the diameter of the BB2 model (D) is defined as the non-dimensional diving depth (as shown in Figure 2), that is, H* = H/D; the non-causal speed is expressed by the Froude number, that is, Fr = $V/\sqrt{gL}$. Where V is the velocity of the underwater vehicle, L is the length of the BB2 model, and g is the gravitational acceleration with a value of 9.81 m/s².

3. Numerical Method

3.1. Control Equation and Turbulence Model

In this paper, the hydrodynamic performance of the optimized BB2 vehicle model was numerically simulated and performed with the help of STAR-CCM+ (18.04.008-R8), and with the RANS equation as the basic equation, the specific form [13] is as follows:

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho u_i\right)+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho u_i{u}_j\right)=-{\displaystyle \frac{\partial p}{\partial x_i}}+\rho f_i+{\displaystyle \frac{\partial }{\partial x_j}}\left[{\mu }_0\left({\displaystyle \frac{{\partial u}_i}{\partial x_j}}+{\displaystyle \frac{{\partial u}_j}{\partial x_i}}\right)-{\displaystyle \frac{2}{3}}{\mu }_0{\displaystyle \frac{{\partial u}_i}{\partial x_i}}{\delta }_{ij}\right]+{\displaystyle \frac{\partial }{\partial x_j}}(-\rho \overline{u_i’u_j’})$$

(1)

where ρ is the fluid density, μ is the fluid viscosity, p is the static pressure, f_i is the mass force per unit mass, and u_i and u_j are the velocity components.

The turbulence model was the SST k−ω model, whose mathematical expression [13] is as follows:

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho k\right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho k{u}_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left({\Gamma }_k{\displaystyle \frac{{\partial }_k}{\partial x_j}}\right)+G_k-Y_k+S_k$$

(2)

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho \omega \right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho \omega u_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left({\mathrm{\Gamma }}_{\omega }{\displaystyle \frac{{\partial }_{\omega }}{\partial x_j}}\right)+G_{\omega }-Y_k+S_{\omega }$$

(3)

where Γ_k and Γ_ω are the effective diffusivities of k and ω, G_k is the turbulent kinetic energy due to the average velocity gradient, G_ω is the generation of special turbulent kinetic energy dissipation, Y_k and Y_ω are the dissipation due to turbulence k and ω, and S_k and S_ω are user-defined source terms.

3.2. Free Surface Treatment

The VOF model [29] can be used to capture the deformation of the free surface and is mainly used in analyzing two or more non-mixing fluids. The transport equation chosen had the form shown in Equation (4) [28]:

$${\displaystyle \frac{\partial C}{\partial t}}+u_s·\nabla C=0$$

(4)

where C is the volume fraction of the two-phase flow, which is equal to the ratio of the fluid volumes, and u_s is the velocity field of the fluid. In this study, C = 1 denotes only water, C = 0 denotes only air, and 0 < C < 1 represents the water–air interface.

3.3. Computational Domain and Boundary Conditions

Under different operating conditions, different computational domain models are selected for calculation. The computational domain of the navigation of the BB2 model at the infinite driving depth is shown in Figure 3a; Figure 3b shows the calculation domain settings for the BB2 model under near-free surface navigation conditions, while Figure 3c shows the calculation domain settings for the BB2 model under near ice surface navigation conditions. As shown in Figure 3, in different computational domains, we refer to the front end of the underwater vehicle as an “Inlet”, the rear end of the underwater vehicle as an “outlet”, the left and right sides of the underwater vehicle as “Side” and “Symmetry”, and the upper and lower sides of the underwater vehicle as “Top” and “Bottom”, respectively.

The boundary conditions we define also vary in different computational domains. In the infinite depth navigation condition, “Inlet, Side, Symmetry, Top and Bottom” are defined as the “Velocity inlet”, and the velocity at the “Velocity inlet” is set as a constant to represent the velocity of the underwater vehicle, while “Outlet” is defined as the “Pressure outlet”; In the near ice surface condition, “Inlet, Side, Symmetry and Bottom” are defined as the “Velocity inlet”, the velocity at the“Velocity inlet” is set as a constant, and “Top” is set as the nonslip wall to represent the ice surface; In conditions of infinite depth and near ice surface, the pressure outlet is set to 0 Pa. In the near free surface navigation condition, “Inlet, Side, Symmetry, Top and Bottom” are defined as the “Velocity inlet”, “Outlet” is defined as the “Pressure outlet”, and the velocity at the “Velocity inlet” is set as a field function: “flat VOF wave along the X-direction”, “Pressure outlet” is set as a field function: “flat VOF wave hydrodynamic pressure”. The setting of boundary conditions under different computational domains is shown in

Table 2. Boundary setting under different calculation conditions.

Boundary Name	Boundary Conditions
	Infinite Depth	Near Free Surface	Near Ice Surface
Inlet/Bottom Side/Symmetry	Velocity inlet	Velocity inlet, velocity based on flat VOF wave along the X-direction	Velocity inlet
Outlet	Pressure outlet	Pressure outlet, based on flat VOF wave hydrodynamic pressure	Pressure outlet
Wall	No slip, impenetrable and fixed wall	No slip, impenetrable and fixed wall	No slip, impenetrable and fixed wall
Top	Velocity inlet	Velocity inlet, velocity based on flat VOF wave along the X-direction	No slip, impenetrable wall with X-direction velocity (ice surface)

.

3.4. Time Step and Time Scale

The most important influencing factor for the selection principle and convergence basis of the fluid time step is the convective Courant number condition, also known as the CFL condition, which is an important concept in the stability and convergence analysis of finite volume methods. The speed at which physical disturbances propagate in the difference equation should be slower than the speed at which time progresses to solve it in order to capture all physical disturbances. For time advancement techniques, there are generally unified time steps and local time steps. Local time steps are generally chosen to accelerate convergence and save computation time.

For the unsteady CFD calculation of vehicle-water interaction, it is necessary to meet the conditional requirements of free surface convection Courant number, which is defined as follows [30]:

$$CFL={\displaystyle \frac{V∆t}{∆x}}$$

(5)

where CFL is the Coulomb number, V is the velocity of the underwater vehicle, ∆t is a time step time, and ∆x is the grid size interval of the region of concern for the Coulomb number.

For the CFD calculation process with a free liquid surface, the maximum time step can be set by the target Coulomb number CFL = 1, that is [30]:

$$CFL={\displaystyle \frac{V∆t}{∆x}}\approx 1$$

(6)

In this paper, the BB2 model (without a propeller) is analyzed under various computational conditions. The time step is determined using Equation (6), resulting in a value of 0.02 s. The total duration of the computation extends for an additional 10 s following the stabilization of the flow field.

In the calculation of near ice surface and infinite depth, the result converges faster because there is no influence of free surface wave; therefore, the total calculation time under the calculation condition near the ice surface and the calculation condition of infinite depth is 30 s, the total calculation time under the calculation condition near the free surface is 40 s. Under different calculation conditions of the BB2 model (with a propeller), different time steps are also adopted, and the calculation is carried out in two parts. The first part is carried out when the propeller does not rotate when the rotation domain is static and the time step is 0.02 s; when the vehicle resistance reaches stability, the rotation speed is assigned to the rotation domain. The time step is changed to the time required for the propeller to rotate 1.65° [31] and continued to be calculated until the flow field stabilizes. At this time, the total calculation time under the calculation condition near the ice surface and the calculation condition of infinite depth is 35 s, and the total calculation time under the calculation condition near the free surface is 45 s.

3.5. Grid Convergence Analysis and Results Verification

In this paper, the computational domains of the model are divided by STAR-CCM+ (18.04.008-R8) cutting-cell mesh, where the important areas that affect the calculation results are refined, and Prismatic mesh is used on the surface of the BB2 model. The total resistance is calculated at a distance of 0.679 m from the free surface, with the set velocity established at 1.61 m/s. Additionally, a grid-independence verification is conducted. By adjusting the grid base size, three groups of different numbers of grids are generated, as shown in

Table 3. Comparison of resistance.

Velocity [m/s]	Basic Size	Number of Meshes	Experimental Results [N]	Numerical Results [N]	Error
1.61	0.05	coarse	4.516	4.612	2.12%
	0.03	medium	4.516	4.57	1.20%
	0.01	fine	4.516	4.515	−0.02%

. The grid base size is set to 0.05, 0.03, and 0.01, respectively; the total number of grids is 2.6 million, 4.232 million, and 7.125 million, respectively. Roy et al. [32] proposed a method for evaluating the uniformity of Cartesian mesh refinement. Where r_G is considered as follows:

$$r_G={\left({\displaystyle \frac{N_{fine}}{N_{coarse}}}\right)}^{(1/d)}$$

(7)

where the total number of grids is denoted as N, and d is the dimension of the computational problem, whose value is 3 in this study. The mesh refinement factors of N_fine N_medium and N_medium/N_coarse are 1.18 and 1.19, respectively. The mesh refinement ratio of N_fine/N_medium (N_medium/N_coarse) is about 1.2, which is consistent with the research results of Huang et al. [33]. We compare the numerical results with the experimental values of Dawson et al. [34], as shown in Table 3. The simulated resistance values calculated under three different grid configurations align closely with the experimental values, exhibiting an error margin of less than 3%. This finding validates the accuracy and feasibility of the proposed numerical simulation method.

In order to evaluate the convergence of the above three grid schemes, and on the basis of the existing research results [35,36], a grid convergence calculation process is prepared, where the grid convergence coefficient R_G is defined as:

$$R_G={\displaystyle \frac{S_3-S_2}{S_2-S_1}}$$

(8)

where S_i is the calculation results of different grids, and i = 1, 2, 3 represent coarse grids, medium grids, and fine grids, respectively; then, by reference to the research of Roache [37], the generalized Richardson extrapolation method is used to evaluate the grid uncertainty, and h_i is defined as:

$$h_i=\sqrt[3]{{\displaystyle \frac{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l} \mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}}{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l} \mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r} \mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l} \mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d} \mathrm{i}}}}$$

(9)

The grid refinement factor [25] is defined as:

$$r_{12}={\displaystyle \frac{h_1}{h_2}}\mathrm{o}\mathrm{r} r_{23}={\displaystyle \frac{h_2}{h_3}}$$

(10)

The discrete order of the final grid [25] is defined as:

$$P_G={\displaystyle \frac{1}{\ln r_{23}}}\left[\ln \left({\displaystyle \frac{1}{R_G}}\right)+\ln \left({\displaystyle \frac{{r_{23}}^{R_G}-1}{{r_{12}}^{R_G}-1}}\right)\right]$$

(11)

$$U_G=1.25\left|{\displaystyle \frac{{\epsilon }_{G32}}{S_2({r_{23}}^{R_G}-1)}}\right|$$

(12)

In Equation (11), when P_G is greater than 2, U_G is calculated with P_G = 2. In Equation (12), ε_G32 is the difference between the calculated resistance R of the fine grids and the medium grids (S₃ − S₂). After analyzing the grid uncertainty of data presented in Table 3, the evaluation factors for grid uncertainty have been determined, as shown in

Table 4. Grid uncertainty evaluation factor.

Variables	RG	PG	UG
Resistance	1.31	1.12	0.0038

. It is evident that the value of U_G is very small, confirming that the numerical simulation results are not sensitive to the spatial resolution used in this study. Since medium grids capture the field more accurately than coarse grids while also providing faster computational speed, reduced resource consumption, and higher efficiency, medium grids have been employed in the calculations conducted in this analysis.

The computational domain grid of BB2 under different conditions is shown in Figure 4. Figure 4a is the refinement of the grids around the hull at infinite depth; Figure 4b is the refinement of the grids around the hull and the free surface; Figure 4c is the refinement of the ice surface grid; Figure 4d–f are the refinement of the grids of the head appendages, hull, and tail, respectively.

4. Comparative Analysis with or Without Appendages

Most scholars’ research on underwater vehicles is carried out on bare hulls without appendages; however, the appendages will cause disturbances to the flow field around the vehicle and threaten safety, especially when sailing in extreme conditions. Shariati et al. [12] conducted research on the impact of appendages on the hydrodynamic characteristics of underwater vehicles operating near the free surface. By comparing the resistance coefficients of the bare hull and appended hull at various Froude numbers and dimensionless depths, they found that the appendages increased the total resistance by approximately 6% compared with the bare hull. In order to investigate the impact of appendages on the hydrodynamic characteristics of BB2 vehicles navigating near a continuous ice layer, a decision regarding the necessity of further research on appended hulls is made by comparing the variations in resistance and lift coefficients of both the bare hull and the appended hull at different Froude numbers. This comparison is conducted while navigating at a distance of 0.6 D from the ice surface.

The expressions for resistance coefficient C_t and lift coefficient C_l [34] are as follows:

$$C_t={\displaystyle \frac{R}{0.5\rho V^{2}S}}$$

(13)

where R is the resistance of the hull, V is the velocity of the underwater vehicle, and S is the wet surface area.

$$C_l={\displaystyle \frac{R_l}{0.5\rho V^{2}S}}$$

(14)

where R_l is the lift of the hull.

The changes in resistance coefficients C_t and lift coefficients C_l of bare hull and appended hull are shown in

Table 5. Comparison of calculation results with and without appendages.

H	Fr	With Appendages		Without Appendages
		Ct × 10−3	Cl × 10−3	Ct × 10−3	Cl × 10−3
0.6 D	0.15	5.19	7.63	6.9	4.1
	0.2250	4.87	8.02	3.74	7.23
	0.397	4.58	9.56	3.48	7.58
	0.45	4.56	9.66	3.46	7.63
	0.552	4.48	9.88	3.57	7.81

below.

In order to observe the changes in the resistance and lift of hulls with and without appendages more intuitively, data in Table 5 are plotted; Figure 5a,b show the comparison of the resistance coefficients and lift coefficients of hulls with and without appendages, respectively. Obviously, with an increase in the Froude number, there are some differences in the resistance coefficients and lift coefficients of the hulls with and without appendages.

Figure 6 shows the dimensionless velocity field of the tail of the hulls with and without appendages at different planes at a velocity of 1.61 m/s (Fr = 0.397) at infinite depth. The X-axis velocity of the vehicle is dimensionless, and V_X/V₀ is obtained, where V_X represents the velocity change in the vehicle along the X-axis and V₀ represents the initial velocity of the vehicle. At infinite depth, the shape distribution of the wakefield of a hull without appendages is approximately round (marked as Round shape), while the shape distribution of the wakefield of the hull with appendages is irregular. Near the appendages, the shape distribution of the wakefield extends from the contours of the tail and the appendages. As the distance from the hull’s tail increases, the shape of the wakefield becomes increasingly uneven (marked as an uneven shape). This indicates that, at infinite depth, the influence of appendages on the hull’s wakefield is significant. It is anticipated that when the vehicle navigates close to the ice surface or the free surface, the disturbance of the flow field around the hull will be more pronounced, making the impact of the appendages on the hull’s wakefield even more substantial. Consequently, the BB2 model with appendages is utilized in this research.

5. Calculation of Operating Conditions and Analysis of the Results (Without a Propeller)

In this section, the BB2 model without a propeller is numerically simulated under various sailing conditions. The characteristics of resistance coefficients and the flow field variations under these conditions are analyzed in detail, providing valuable insights into the vehicle’s performance in real-world environments. The model utilized in the numerical simulation is illustrated in Figure 1.

5.1. Resistance Coefficient Results and Analysis

Figure 7 illustrates the variation in the resistance coefficient (C_t) of the BB2 vehicle in relation to the Froude number (Fr) under various operating conditions. At this point, the distance between the vehicle and the free surface or ice surface is 0.6 D, while infinite depth indicates that the vehicle is at an infinite distance from the free surface or ice surface. The formula for C_t is presented in Equation (13).

Figure 7 illustrates that when BB2 is positioned 0.6D away from the free surface, the resistance coefficients initially increase and then decrease as the Froude number (Fr) varies from 0.15 to 0.76. In conditions of near continuous ice layer navigation and infinite depth navigation, the fluctuations in the resistance coefficients of the BB2 vehicle are less pronounced compared with those observed during near-free surface navigation. Overall, this indicates a gradual decline followed by a stabilization trend.

Figure 8 illustrates the variation in the resistance coefficient with dimensionless depth (H*) at a constant velocity of 1.61 m/s. These data indicate that the resistance coefficient of the vehicle fluctuates significantly under different operating conditions when the distance between the vehicle and the free surface/ice surface is minimal; however, the difference in the resistance coefficient between the near-ice surface and the vehicle at infinite depth is not substantial. Under conditions near the free surface, the resistance coefficient of the vehicle exhibits considerable variation as the distance from the free surface increases. As the dimensionless depth (H*) near the free surface increases, the resistance coefficient of the vehicle gradually decreases. When the dimensionless depth (H*) of the vehicle from the free surface or ice surface exceeds 3, the resistance coefficient remains nearly constant across different operating conditions. Furthermore, the free surface has a more pronounced effect on the resistance coefficient of the underwater vehicle compared with the ice surface.

Upon observing Figure 8 and Figure 9, it is evident that the influence of the free liquid surface on the vehicle’s resistance coefficient is more pronounced compared with that of continuous ice layers. This is attributable to the wave-making effect of the free liquid surface, which intensifies the interaction between the vehicle and the water. In contrast, the presence of continuous ice layers tends to suppress this wave-making effect, resulting in a less intense vehicle-water interaction; however, an analysis based solely on the resistance coefficient is insufficiently persuasive; therefore, we will conduct further research on the flow field of the vehicle under various navigation conditions in subsequent chapters.

5.2. Velocity Field Analysis

Figure 9 shows the velocity field distribution of the BB2 vehicle under various operating conditions, with a set velocity of 1.82 m/s. At infinite depth (Figure 9a), the velocity field distribution around the vehicle is relatively uniform and similar to the axisymmetric distribution. Near the free surface (Figure 9b), the asymmetry of the flow field distribution around the vehicle is further exacerbated. This asymmetry arises from the wave-making disturbances caused by the vehicle interacting with the free surface, resulting in peaks (BF) at the head and troughs (BG) at the tail of the vehicle. The peaks lead to an increase in pressure at the head, while the troughs result in a decrease in pressure at the tail, creating a dynamic pressure difference between the head and tail. This dynamic pressure difference, induced by wave-making, generates resistance that contributes to a more disordered velocity field distribution around the vehicle, forming a reflux zone (V_h − 1) at the tail. When navigating near the ice surface, the velocity field around the vehicle remains asymmetrically distributed, and a reflux zone (V_h − 2) appears at the tail. Additionally, due to the interaction of the wakefield with the continuous ice layer, the low-velocity zone (V-low3) of the wakefield near the continuous ice layer (Figure 9c) is larger than the low-velocity zone (V-low1) of the wakefield at an infinite depth and the low-velocity zone (V-low2) of the wakefield near the free surface. The overall distribution is uneven, with most distribution above the tail.

Figure 10 shows the distribution of the dimensionless velocity field at different velocities, measured at a distance of H = 0.6D from the ice surface. Due to the proximity to the ice surface, the flow field around the tail of the vehicle is significantly influenced by the ice, resulting in the formation of a recirculation zone near the tail and a low-velocity region at the rear. In contrast, the flow field near the front of the vehicle remains largely unchanged. With an increase in velocity, the ice surface has little influence on the flow field near the head, and the adsorption effect on the wakefield of the vehicle is weakened. The size of the tail reflux area follows the order: V_h − 1 > V_h − 2 > V_h − 3 > V_h − 4 > V_h − 5. Furthermore, as the velocity increases, the low-velocity zone at the tail of the vehicle gradually decreases.

In order to directly observe the adsorption effect of a continuous ice layer on the tail of the vehicle, a cross-section was taken at 1.674 m from the front of the vehicle to analyze the changes in the dimensionless velocity field at this location. The position of the intercepted cross-section is illustrated in Figure 11.

Figure 12a–e show the X-direction velocity at a distance of 1.674 m from the head of the submarine, with velocities of 0.609 m/s, 0.913 m/s, 1.61 m/s, 1.82 m/s, and 2.61 m/s, respectively. It is clear that there is a reflux zone (V_h) and a low-velocity zone (V-low) in the tail of the vehicle, as well as a wakefield caused by the tail wing appendages. When the velocity changes from V = 0.609 m/s to V = 2.61 m/s, the amplitude and range of the reflux zone (V_h) and the low-velocity zone (V-low) are reduced, and the adhesion between the wakefield caused by tail wing appendages and the flow field of appendages at the ice layer gradually weakens, which once again verifies that the adsorption effect of the ice surface decreases with increasing velocity.

Figure 13a–f show the dimensionless velocity field distribution when H is 0.6D, 1D, 1.5D, 2D, and 3D near the continuous ice layer and at infinite depth. It can be seen that when H = 0.6D and 1D near the ice surface, due to the influence of the continuous ice layer, the velocity field around the vehicle is asymmetrically distributed; the ice has a significant adsorption effect on the velocity field of the head and tail of the vehicle. At the vehicle’s tail, there are distinct zones: a reflux zone (V_h − 1 and V_h − 2), a low-velocity zone (V-low1, V-low2), and a high-velocity zone (V-high1 and V-high2). When H ≥ 1 D, the low-velocity zone (V-low2, V-low3, V-low4, V-low5, and V-low6) at the tail of the vehicle tends to become symmetrically distributed, while the high-velocity zone also exhibits symmetrical distribution without significant changes. When H ≥ 1.5D, the reflux zone at the tail of the vehicle disappears. When H ≥ 2D, the impact of the continuous ice layer on the velocity field of the head and tail disappears, and the velocity field of the head of the vehicle tends to be symmetrically distributed and is consistent with the velocity field distribution of the head at infinite depth. When H = 2D, the adsorption effect of the continuous ice layer on the velocity field of the tail of the vehicle completely disappears, but there is still an impact on the central velocity field of the vehicle (V-mid4), and the central velocity field distribution of the vehicle tends to be the same as that at infinite depth (V-mid6). When H = 3D, the velocity field distribution around the vehicle is already almost the same as the Figure 13f (Infinite depth).

In order to more intuitively represent the influence of the free surface on the vehicle during navigation near the free surface, Figure 14 shows the dimensionless velocity field distribution near the free surface at different depths, and the velocity is set to 1.61 m/s. It can be seen that when H = 0.6D and 1D near the free surface, the flow field around the vehicle changes significantly; the velocity field around the vehicle is asymmetrically distributed; the velocity field at the tail of the vehicle is consistent with the wave-making trend of the free surface and similar to the navigation near the continuous ice layer, there is the formation of reflux zone (V_h − 1, V_h − 2), a low-velocity zone (V-low1, V-low2) and a high-velocity zone (V-high1, V-high2) at the tail of the vehicle. When the vehicle is navigating near the free surface, there is obvious wave-making on the free surface, which forms peaks (BF) above the rear ends of both the head and tail of the vehicle and troughs (BG) at the middle of the vehicle, respectively.

Dissimilar to navigation near the ice surface, navigation near the free surface exhibits distinct characteristics. When H ≥ 2D, the low-velocity zone at the tail of the vehicle (V-low4, V-low5, V-low6, V-low7) tends to be symmetrically distributed, the high-velocity zone is symmetrically distributed, and the reflux zone of the tail almost disappears. When H ≥ 3D, the impact on the vehicle wakefield can be almost negligible. The velocity field of the head tends towards a symmetrical distribution, and the overall flow field shows a tendency towards a symmetrical distribution. When H = 3D, the free surface tends to flatten, and the wave-making effect decreases sharply. It can be seen that the free surface has little effect on the resistance of the vehicle from Figure 8. When H = 5D, the influence of the free liquid surface on the velocity field of the vehicle completely disappears.

Figure 15 shows the wave height variation in the free surface when the vehicle navigates near the free surface at different depths. It can be seen that when the navigation depth is less than 1D, there are obvious peaks of the free surface behind the tail, as well as noticeable fan-shaped waves. When the navigation depth continues to increase, the fan-shaped waves gradually weaken until the impact of the free surface completely disappears when H = 5D, which is consistent with the law of change at the free surface shown in Figure 14.

By comparing Figure 13 and Figure 14 and incorporating Figure 15, it is evident that the wave-generating effect of the free liquid surface significantly influences the flow field around the vehicle. The vehicle-water interaction is pronounced, while the continuous ice layer exhibits a certain inhibitory effect on the wave-generating capability of the free liquid surface. This observation supports the hypothesis presented in Section 5.1.

5.3. Pressure Field

Figure 16 shows the variation in the dimensionless pressure field on the surface of the vehicle under the influence of ice surface at different depths of navigation. The definition of the pressure coefficient [17] is as follows:

$$C_P={\displaystyle \frac{P_i}{0.5\rho V^{2}S}}$$

(15)

where i =1 or 2, P₁ represents the total pressure experienced by the aircraft when sailing near the ice surface because in the near-ice navigation condition, the static water wave effect is not considered, and the total pressure experienced by the vehicle at this time is the dynamic pressure. P₂ represents the dynamic pressure exerted on the vehicle during near-free surface navigation, as the conditions of near-free surface navigation are affected by static water waves. At this point, P₂ is the total pressure minus the static water pressure.

Obviously, with an increase in the distance near the ice surface, the range and amplitude of the negative pressure zone (P_F) at the head and tail of the vehicle are becoming smaller. When H ≥ 2D, the pressure distribution of the vehicle hardly changes. At this time, the ice surface has less influence on the vehicle.

Figure 17 shows the variation in the dimensionless pressure field on the surface of the vehicle, influenced by the free liquid surface at various depths of navigation. Similar to the navigation near the ice surface, when the distance of the vehicle from the free liquid surface changes from 0.6D to 5D, the range and amplitude of the negative pressure zone (P_F) at the head and tail of the vehicle becomes smaller. When the vehicle is 0.6 D, 0.8 D, 1 D, and 1.5D away from the free surface, the influence of the free surface on the surface pressure of the vehicle is more significant than that of the ice surface. When the vehicle is 0.6D and 0.8D away from the free surface, there is a significant positive pressure zone (P_Z − 1 and P_Z − 2) above its head and a significant negative pressure zone (P_F − 9 and P_F − 10) above its tail. When the BB2 vehicle is navigating near the free surface, the peaks (BF) of the head of the vehicle cause an increase in the pressure above the head, while the troughs (BG) near the tail cause a decrease in the pressure, which results in a dynamic pressure difference of fluid between the head and the tail and causes the resistance of waving making. When H ≥ 3D, the pressure distribution of the vehicle is similar to that on the surface of the vehicle during navigation near the ice surface and slowly approaches the surface pressure distribution during navigation at infinite depth.

Figure 18a shows the variation in the dimensionless pressure field along the X-axis direction on the surface of the vehicle influenced by the ice surface at various navigation depths. Figure 18b depicts the changes in dimensionless pressure along the X-axis on the vehicle’s surface under the influence of the free liquid surface at different navigation depths. During navigation near the ice surface, the dimensionless pressure field distribution of the vehicle in the X direction at different diving depths is roughly the same, with its variation law consistent with that in Figure 16. It can be seen that during navigation near the free surface, when the navigation depth is less than 1D, the pressure distribution near the head and tail of the vehicle is significantly different from that at other depths, with its variation law consistent with that in Figure 17.

6. Comparison of Different Operating Conditions in Self-Propulsion

6.1. Model of Self-Propulsion Condition

In the above sections, the influence of free surface and ice surface on the resistance, velocity field, and pressure field of the appended BB2 model is discussed without taking into account the models with a propeller. Since the propeller is an indispensable part of powering a vehicle during its navigation, the self-propulsion of a vehicle with a propeller under different operating conditions should be studied; therefore, Section 6 and Section 7 studied the self-propulsion of underwater vehicles under different operating conditions at infinite depth, near free surfaces, and near ice surfaces, using a three-dimensional model, as shown in Figure 19.

The model (Figure 19a) in self-propulsion being studied is a BB2 model (Figure 19b) appended with a 6-blade propeller (Figure 19c). The parameters of the BB2 model with appendages in Figure 19b are the same as the parameters in Table 1 in Section 1.

The specific parameters of the model with a 6-blade propeller are shown in

Table 6. 6 Specific parameters of blade propeller model.

Items	Geometric Parameter	Unit
blades	6	piece
Dpro	0.118	m
Dhub/Dpro	0.2	-
AE/AO	0.80	-
P0.7Rpro	0.997	-

. where D_pro is the propeller diameter, D_hub is the hub diameter, D_hub/D_pro is the ratio of the hub to diameter, and P_0.7Rpro is the pitch at 0.7 times the propeller radius.

6.2. Computational Domains and Meshing at Self-Propulsion Condition

The division of the computational domain under self-propulsion conditions aligns with Figure 3 in Section 3.3. The boundary conditions for the computational domain across various operational scenarios are consistent with the parameters outlined in Table 2 of Section 3.3. In the self-propulsion computational scenario, the BB2 model without a propeller is substituted with the BB2 model that includes a propeller, and an additional rotating domain is incorporated. For instance, the self-propulsion computational domain near the ice surface is illustrated in Figure 20.

As shown in Figure 20, for self-propulsion near the ice surface, simply replace the BB2 model with appendages in Section 2 with the BB2 self-propulsion of the 3D model (Figure 19). In the numerical simulation process, we set up a cylindrical component containing a propeller, as shown in Figure 20, and named the region obtained by performing Boolean operations and subtraction operations on this component and the BB2 self-propulsion 3D model as the rotation domain. We also performed Boolean operations and subtraction operations on the overall calculation domain, rotation domain, and BB2 self-propulsion 3D model and named the obtained region as the calm water domain. We created an interface between the rotation domain and the calm water domain for data exchange. In the process of simulating the self-propulsion of the vehicle, it is only necessary to apply a certain flow velocity, give rotation motion to the rotation domain, and adjust the rotation rate of the rotation domain according to the requirements. This is the rotation rate of the propeller. The sliding grid is used for data exchange between the propeller rotation domain and the static water domain. Similarly, for navigation at infinite depth and near the free surface, the appended BB2 model in Section 2 is replaced with the BB2 self-propulsion model, and a sliding grid is used for data exchange between the propeller rotation domain and the static water domain. The remaining parameters of the computational domain are the same as the parameters in Table 2 in Section 3.3.

The meshing of the BB2 vehicle model under different self-propulsion conditions is shown in Figure 21. The meshing of the hull of the vehicle and its surroundings under different self-propulsion conditions is the same. Figure 21a is the meshing around the vehicle at infinite depth; Figure 21b is the refined overall meshing of the area around the hull and of the free surface during navigation near the free surface; Figure 21c is the meshing of the boundary layer of the head of the submarine; Figure 21d is the refined meshing of the ice surface during navigation near the ice surface; Figure 21e–g are the meshing of the rear propeller, the refinement of the propeller blades, and the overall meshing of the blade’s boundary layer.

6.3. Meshing Sensitivity Study Under Self-Propulsion Condition

Please refer to Section 3.5 for the process of grid-independent verification and uncertainty analysis under self-propulsion conditions, which utilizes the three-dimensional model shown in Figure 19a. On the basis of meshing in Section 3.5, only the meshing of the rotation domain is added. By reference to the research of Pan et al. [38], the following self-propulsion point calculation process is formulated: First, the self-propulsion calculation is performed at least three different velocities to obtain the corresponding propeller thrust and hull resistance, followed by a drawing of a change curve of the propeller thrust and hull resistance with the change in velocity; then, the velocity corresponding to the intersection of the two curves and the propeller velocity of the self-propulsion model at the corresponding velocity, that is, the self-propulsion point are found. Taking the self-propulsion of a vehicle at an infinite depth with a flow velocity of 1.22 m/s as an example, the propeller thrust and hull resistance at rotational speeds of 8 rps, 10 rps, and 12 rps are calculated and plotted, as shown in Figure 22. It can be calculated that the self-propulsion velocity at a flow velocity of 1.22 m/s at infinite depth is 10.46 rps.

The use of the three-dimensional model is shown in Figure 19a. Taking the flow velocity of 1.22 m/s and a rotational speed of 10.46 rps at infinite depth as an example, by adjusting the basic dimensions of the static water domain and the rotational domain during meshing, three sets of grids of different densities under self-propulsion conditions are generated, namely coarse-1, medium-1 and fine-1 for grid independence verification, where the total number of grids is 3.68 million, 5.36 million, and 8.57 million respectively; the calculation process of grid convergence under self-propulsion conditions is the same as that in Section 3.5.

The calculation results of the hull resistance (R) and propeller thrust (T) under different grid numbers are shown in

Table 7. The calculation results under different grid numbers of self-propulsion operating conditions.

Velocity [m/s]	Number of Meshes	R [N]	T [N]
1.61	coarse-1	3.97	4.40
	medium-1	3.92	4.34
	fine-1	3.88	4.30

.

After analyzing the grid uncertainty of data in Table 7, the results of the evaluation factors of the grid uncertainty are obtained, as shown in

Table 8. Grid uncertainty evaluation factor.

Variables	RG	PG	UG
R	0.8	2	0.0017
T	0.67	2	0.0013

. It should be noted that when the P_G calculated by formula (11) in Section 3.5 is greater than 2, P_G = 2 is taken for the calculation of U_G.

It can be seen that the value of U_G is very small, which confirms that the numerical simulation results are not sensitive to the spatial resolution herein. Since medium grid-1 is finer than coarse grid-1 in capturing the flow field under self-propulsion conditions, the computation speed is faster than that of fine grid-1, which takes up less resources and achieves high efficiency; therefore, medium grid-1 is adopted for the calculations under self-propulsion conditions.

7. Calculations Conditions and Result Analysis Under Self-Propulsion Condition

In this section, numerical simulation is performed for the vehicle at the same velocity and different depths at the free surface, a distance of H = 0.6D, 1D, 2D from the near ice surface, and during self-propulsion at infinite depth. First, according to the method of finding the velocity of the self-propulsion point in Section 6.3, three different velocities are selected for each operating condition for numerical simulation, thereby obtaining the velocity of the self-propulsion point under different self-propulsion conditions. At the same time, a numerical simulation of the operating conditions at the self-propulsion point is conducted, followed by a comparative analysis of the self-propulsion velocity, velocity field, and pressure field under the coupling effects between the vehicle near the free surface and the free surface itself.

7.1. Rotational Speed at Self-Propulsion Point Under Different Self-Propulsion Conditions

In order to determine the self-propulsion point of the vehicle, numerical simulations and calculations are conducted to assess the hull resistance and the hydrodynamic performance of the propeller under the following conditions: a fixed velocity of V = 1.22 m/s, a depth of H = 0.6D from the free surface, and propeller rotational speeds of n = 13 rps, 15 rps, and 17 rps. Here, n denotes the rotational speed of the propeller. When the vehicle is positioned at a depth of H = 1D from the free surface, numerical simulations and calculations are conducted at propeller rotational speeds of n = 9 rps, 11 rps, and 13 rps. Conversely, when the vehicle is at a depth of H = 2D from the free surface, or at depths of H = 0.6D, 1D, and 2D from the ice surface, or at infinite depth, numerical simulations and calculations are performed to assess hull resistance and the hydrodynamic performance of the propeller at rotational speeds of n = 8 rps, 10 rps, and 12 rps. The resistance experienced by the hull and the thrust generated by the propeller at various velocities are recorded to determine the rotational speed at the self-propulsion point under different self-propulsion conditions. The specific calculation results are presented in

Table 9. The calculation results under different self-propulsion conditions.

H	Conditions	Rotational Speed [rps]	T [N]	R [N]	Rotational Speed at Self-Propulsion Point [rps]
0.6D	Free surface	13	8.83	10.5	13.94
		15	12.96	11.07
		17	17.93	11.85
	Ice surface	8	1.46	4.45	10.77
		10	3.89	4.9
		12	7.02	5.42
1D	Free surface	9	2.76	6.32	11.95
		11	5.47	6.89
		13	8.94	7.36
	Ice surface	8	1.37	4.11	10.6
		10	3.79	4.57
		12	6.89	5.07
2D	Free surface	8	2.2	4.93	10.46
		10	4.6	5.21
		12	7.72	5.7
	Ice surface	8	1.379	3.92	10.450
		10	3.82	4.38
		12	6.88	4.95
∞	Infinite	8	1.379	3.92	10.459

.

From Table 9, it can be observed that during navigation near the free surface, the rotational speed at the self-propulsion point varies significantly at different distances from the free surface. In contrast, when navigating near the ice surface, the rotational speed at the self-propulsion point changes very little, and when H = 2D near the ice surface, the rotational speed at the self-propulsion point is almost the same as that at infinite depth. As can be seen from Section 5, the free surface causes greater flow field disturbance on the vehicle than the ice surface; as a result, the resistance experienced by the hull is greater than that encountered near the continuous ice layer. In self-propulsion, however, the irregular flow field disturbance caused by the free surface results in certain energy dissipation of the propeller during the generation of thrust; therefore, at the same velocity and the same navigation depth, self-propulsion near the free surface requires a higher rotational speed of the propeller to generate sufficient thrust. Hence, the closer the vehicle is to the free surface, the higher the rotational speed at the self-propulsion point. From Section 3, it can be seen that the influence of the ice surface on the vehicle is relatively small, and when H ≥ 2D, the influence of the resistance coefficient of the ice surface on BB2 can be ignored; therefore, when H = 2D from the ice surface, the rotational speed at the self-propulsion point is almost the same as that at infinite depth.

7.2. Velocity Field Under Different Self-Propulsion Conditions

Figure 23 shows the velocity field distribution of a BB2 vehicle (with a propeller) under different operating conditions (V = 1.22 m/s). During navigation near the free surface (Figure 23a), the asymmetry of the flow field distribution around the vehicle is further exacerbated. This is due to the wave-making disturbance by the vehicle on the free surface, which causes continuous peaks (BF) and troughs (BG) at its head and tail. The peaks cause an increase in the pressure at the head, while the troughs cause a decrease in the pressure at the tail, which forms the dynamic pressure difference of the fluid between the head and the tail. This dynamic pressure difference caused by wavemaking forms a resistance that makes the velocity field distribution around the vehicle more disordered; however, compared with the presence of a low-velocity zone surrounding the wakefield of the vehicle without a propeller in Section 5.2, an approximately symmetrical transverse high-velocity zone (V-high1) is generated at the wakefield of the vehicle with a propeller due to its rotation. The influence of the free surface on the velocity field in this high-velocity zone is not severe. At a far distance from the propeller, this transverse high-velocity zone is expanding outward. This is because, under the self-propulsion condition, the high rotational speed of the propeller is converted into thrust on the vehicle, which can reduce the impact of wave-making resistance; however, at a distance far from the propeller, as the propeller continuously causes waving making on the free surface, energy dissipation of the thrust of propeller results. During navigation near the ice surface (Figure 23b), the velocity field distribution around the vehicle is also asymmetric. The transverse high-velocity zone at the tail of the vehicle does not exhibit significant diffusion when compared with navigation near the free surface. This observation further demonstrates that the free surface has a more pronounced influence on the vehicle’s velocity field than the continuous ice layer. At infinite depth (Figure 23c), the velocity field distribution around the vehicle is relatively uniform and similar to the axisymmetric distribution, and the high-velocity zone at the tail of the propeller is axisymmetric.

7.3. Pressure Field Under Different Self-Propulsion Conditions

Figure 24 shows the dimensionless pressure field distribution of the BB2 vehicle under different self-propulsion conditions. Figure 24a shows the dimensionless pressure field distribution under the self-propulsion condition where the vehicle is 0.6D near the free surface. Figure 24b shows the dimensionless pressure field distribution under self-propulsion conditions where the vehicle 0.6D is from the ice surface. Figure 24c shows the dimensionless pressure field distribution under self-propulsion conditions where the vehicle is at infinite depth. All calculations were made at a velocity of 1.22 m/s. Due to the waving making to the free surface, the troughs (BG1) cause the negative pressure zone (P_F − 1) at the front end of the vehicle, while the peaks (BF2) cause the positive pressure zone (P_Z − 1), which is consistent with the result in Section 3.3. During the self-propulsion of the BB2 vehicle near the continuous ice layer, the range of the low-pressure zone generated near the head and tail of the hull is greater than that during self-propulsion at the infinite depth. This indicates that the ice surface causes a certain influence on pressure field distribution during vehicles’ self-propulsion, but the influence is less significant compared with that caused by the free surface.

8. Conclusions

This article conducts numerical simulations of the BB2 (without a propeller) model and the BB2 (with a propeller) model under different operating conditions and compares and analyzes the results under different operating conditions. The following conclusions are drawn.

(1)

During navigation near the free surface, the resistance coefficient of the vehicle is significantly greater than that observed during navigation at infinite depth and near the ice surface due to the influence of the free surface. The resistance coefficient decreases as the distance from the ice surface and the free surface increases. When the navigation depth exceeds 2D, the influence of the ice surface on the BB2 vehicle can be disregarded. Additionally, when H ≥ 3D, the influence of the free surface on the BB2 vehicle can also be ignored.

(2)

Regardless of navigation near the continuous ice layer or near the free surface, the velocity field distribution of BB2 remains consistently asymmetric. During navigation near the ice surface, the ice surface has a significant adsorption effect on the wakefield of BB2. As the velocity increases, the adsorption effect gradually weakens. When the navigation depth is greater than 2D, the influence of the ice surface on the velocity field of BB2 gradually weakens.

(3)

During navigation near the continuous ice layer, negative pressure zones are observed at both the head and tail of BB2. The extent of these negative pressure zones gradually decreases with increasing distance from the ice surface. In contrast, when navigating near the free surface, the peaks and troughs of the free surface, there is a positive pressure zone at the tail. The influence of wave generation on the pressure field of BB2 is significantly greater than that of the ice surface.

(4)

During self-propulsion, when the diving depth H = 1D, due to the energy dissipation caused by wave generation at the free surface results in a significantly higher rotational speed at the self-propulsion point compared with that observed during self-propulsion near the ice surface. Consequently, the influence of the free surface on the vehicle’s velocity and pressure fields is marked greater than that of the ice surface.

The limitation of this study is that the influence of open degrees of freedom was not considered in the numerical simulation process, and fixed models were used for numerical simulation calculations. Moreover, with the impact of global warming, Arctic glaciers are melting, and large areas of fragmented ice will appear in the originally flat ice regions. In the future, it is necessary to consider the changes in the hydrodynamic performance of underwater vehicles when navigating near fragmented ice, as well as the impact of collisions between the vehicle and floating fragmented ice on the vehicle.